This randomized phase III trial studies how well transarterial chemoembolization (TACE) works compared to stereotactic body radiation therapy (SBRT) or stereotactic ablative radiation therapy (SABR) in patients with liver cancer that remain after attempts to remove the cancer have been made (residual) or has come back (recurrent). TACE is a minimally invasive, image-guided treatment procedure that uses a catheter to deliver both chemotherapy medication and embolization materials into the blood vessels that lead to the tumors. SBRT or SABR may be able to send radiation directly to the tumor and cause less damage to normal liver tissue. It is not yet known whether TACE is more effective than SBRT or SABR in treating patients with persistent or recurrent liver cancer who have undergone initial TACE.